Yet another Physics Question that I'm unsure about....Please help?

Question

The acceleration function of a bus is given by a(t) = 1.2t

a.) derive the bus's velocity and position functions. Show all appropriate calculus and algebraic steps.

b.) if the bus's velocity at time t = 1.0 sec is 5.0 m/s, what is its velocity at time t = 2.0 sec?

c.) if the bus's position at time t = 1.0 sec is 6.0 m, what is its position at time t = 2.0 sec?

After differenciating, i got
v = 2.4t^2 + vi
x = 4/5t^3 + vit + xi

are these right?, and if they are how would i find the next two answers?

plz show me your steps

Answer 1

> After differenciating, i got
> v = 2.4t^2 + vi
> x = 4/5t^3 + vit + xi

This is wrong. To go from acceleration to velocity to position, you INTEGRATE, not differentiate. And you've done the integration wrong. To integrate a power of t, you add 1 to the exponent, then DIVIDE by the new exponent (looks like you multiplied by the exponent instead).

That takes care of part (a)

> how would i find the next two answers?

To do that, you have to figure out the values of vi and xi. There is enough info in the problem to help you figure that out. Consider:

"...the bus's velocity at time t = 1.0 sec is 5.0 m/s"

That means, take your (corrected) equation for v, then plug in 1.0 sec for t, and set the result equal to 5.0 m/s. That will give you an equation that you can solve to get vi. Once you've got vi, you can plug in t=2.0 sec to answer part (b).

"...if the bus's position at time t = 1.0 sec is 6.0 m"

Take your (corrected) equation for x, plug in 1.0 sec for t, and plug in the vi value that you calculated in part (b), and set the whole thing equal to 6.0 m. That will give you an equation that you can solve to get xi.

Once you've got xi, you can plug in t=2.0 sec to answer part (c).

Answer 2

Your integrals are wrong:

a(t) = 1.2t

v(t) = integral of a(t)dt = 0.6 t^2 + v0

x(t) = integral of v(t)dt = 0.2 t^3 + v0t + x0

They give you v at a specific t, which let's you get v0
v0 = v(t) - 0.6 t^2

Then just plug in to get v at any other t.

They give you x at a specific t, which let's you get x0
x0 = x(t) - 0.2 t^3 - v0t

Then just plug n to get x at any other t.